Micro-Workshop on the Formal Theory of PDEs

November 6-10, 2017, DipMat, University of Salerno

The formal theory of PDEs originated in the first half of the 20^th century from the works of Riquier, Janet, Elie Cartan and a few others. Later, it was recast in modern, differential geometric language by several people including Spencer, Bryant, Chern, Goldschmidt, Griffiths, ... Our tiny and informal meeting aims at comparing different approaches to the formal theory of PDEs within jet spaces. It also aims at discussing the role of the formal theory of PDEs in contemporary differential geometry and mathematical physics. The workshop will consist of three 6 hours mini-courses, one 2 hours lecture, and (hopefully) several hours of open discussion.

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Group Picture

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Speakers

Francesco Cattafi (University of Utrecht)

Marius Crainic (University of Utrecht)

Igor Khavkine (University of Milan)

Ori Yudilevich (Catholic University of Leuven)

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Other Participants

Antonio De Nicola (University of Salerno)

Marco Di Mauro (University of Salerno)

Pier Paolo La Pastina (University of Rome “La Sapienza”)

Antonio Miti (Catholic University of Milan)

Jonas Schnitzer (University of Salerno)

Luca Vitagliano (University of Salerno)

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Schedule

This schedule is tentative and may be subject to (significant) changes!

	Mon 6	Tue 7	Wed 8	Thu 9	Fri 10
09:00 – 09:45	Khavkine 1 (room: DipMat)	--	--	--	--
	break
10:00 – 10:45	Khavkine 1 (room: DipMat)	Yudilevich 2 (room: DipMat)	--	--	Yudilevich 3 (room: DipMat)
	questions	break			break
11:00 – 11:45	Cattafi (room: DipMat)	Yudilevich 2 (room: DipMat)	Crainic 2 (room: DipMat)	Khavkine 3 (room: DipMat)	Yudilevich 3 (room: DipMat)
	break	questions	break	break	questions
12:00 – 12:45	Cattafi (room: DipMat)		Crainic 2 (room: DipMat)	Khavkine 3 (room: DipMat)
	questions		questions	questions
Lunch
14:00 – 14:45	--	Khavkine 2 (room: DipMat)		--	Crainic 3 (room: F3)
		break			break
15:00 – 15:45	--	Khavkine 2 (room: DipMat)		Yudilevich 2 (room: F3)	Crainic 3 (room: F3)
		questions		break	questions
16:00 – 16:45	Crainic 1 (room: F3)	discussions (room: DipMat)		Yudilevich 2 (room: F3)	discussions (room: F3)
	break			questions
17:00 – 17:45	Crainic 1 (room: F3)
	questions

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Titles and Abstracts

Francesco Cattafi (lecture)

Title: PDEs and Pfaffian Bundles

Abstract: I will begin this talk by recalling the notion of jet bundle and how to use it to formalise partial differential equations of order k as submanifolds P in J^kE.

However, the whole jet structure is not really needed: as we will see, the fundamental information of the PDE is encoded in P and in a differential form q called the Cartan form. This new object (P, q) is called Pfaffian bundle and can be defined in a way completely independent from the jet structure it originates from. An equivalent definition uses the language of distributions, generalising what in the previous framework was the Cartan distribution (the kernel of the Cartan form).

With this new and more general formalism, we can understand better other properties of the original equation: for example, if it is linear (and, if not, how to "linearise" it) or if it is integrable (and, if so, how to construct a prolongation sharing the same solutions with the starting equation). Accordingly, we will first formalise these problems in the settings of Pfaffian bundles and then give a brief idea of their solutions.

This is a joint work with Marius Crainic and Maria Amelia Salazar.

Marius Crainic (mini-course)

Title: (Almost) Gamma-structures

Abstract: Geometric structures are usually encoded into a more algebraic object (e.g. a differential form or tensor perhaps satisfying some non degeneracy condition) on which one imposes an "involutivity condition" (usually a PDE that it has to satisfy).

If one gives up on the involutivity condition one talks about "almost geometric structures". The existence of "almost" structures on a given manifold is usually a topological problem (completely controlled by simple algebraic-topological invariants, such as the vanishing of certain cohomology classes); in contrast, the existence of genuine geometric structures is usually a hard analytic problem. For instance, almost symplectic structures on M are no-where non-degenerate 2-forms on M; if the form is closed, we talk about the symplectic structures. Similarly for complex structures, for which the PDEs are encoded in the Nijenhuis tensor of an almost complex structure.

Passing from an almost structure to a genuine one is known as "the integrability problem". One standard framework to make sense of this (and understand many of the known geometric structures in an unified way) is via "the theory of G-structures". In the first day I will give an overview of that theory, moving towards the more general framework of Gamma-structures. However, while the notion of Gamma-structure is standard and well-studied, the corresponding one of "almost Gamma-structures" is much less so (except for the transitive case, which, in principle, is that of G-structures). The aim for lectures 2 and 3 is to explain how to handle the integrability problem in this more general context (but with some gain also for the more classical situations). More precisely, lecture 2 will be devoted to the framework that allows one to handle Lie pseudogroups Gamma: that of Pfaffian groupoids. Then, in the last lecture, I will explain how the notion of Morita equivalence and Morita map, adapted to the Pfaffian context, allows one to treat almost Gamma-structures. 

Igor Khavkine (mini-course)

Title: Topics in the Formal Theory of PDEs

Abstract: We will discuss formal integrability and involutivity of PDEs via Spencer cohomology and its relation to commutative algebra. Time permitting, we will also discuss some applications, like constructing compatibility complexes and counting formal degrees of freedom of solutions.

Ori Yudilevich (mini-course)

Title: Formal Integrability of PDEs, Lie pseudogroups and Cartan-Ehresmann Connections

Abstract: In his pioneering work on Lie pseudogroups, Élie Cartan showed that one can associate a certain set of structure equations with any Lie pseudogroup, and that, in a sense, these equations fully encode the Lie pseudogroup and can be used to study its structure. These equations generalize the well-known Maurer-Cartan equations of a Lie group, and they contain an extra "mysterious" term that involves a certain collection of auxiliary 1-forms. This collection of 1-forms, as it turns out, can be interpreted as a special Ehresmann connection on the defining system of PDEs of the Lie pseudogroup, and we call it a Cartan-Ehresmann connection. 

I will divide my lectures into two parts. In the first part, I will discuss the notion of a Cartan-Ehresmann connection in the generality of jet bundles and PDEs (viewed as submanifolds of jet bundles), and I will show how it can be used to give a simple and rather transparent proof of Goldschmidt's theorem of formal integrability of PDEs. In the second part, I will give a brief overview of Cartan's structure theory for Lie pseudogroups and discuss the role of Cartan-Ehresmann connections in the theory.

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Organizers

Antonio De Nicola (University of Salerno)

Pier Paolo La Pastina (University of Rome “La Sapienza”)

Jonas Schnitzer (University of Salerno)

Luca Vitagliano (University of Salerno)

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More Information

Here is a map with relevant places in Fisciano, inside and outside the campus.

People interested in attending the meeting should send an e-mail to: lvitagliano@unisa.it